#### Please solve RD Sharma class 12 chapter Differential Equations exercise 21.11 question 25 maths textbook solution

Answer: $y^{2}=4 x$

Given: Distance between foot of ordinate of the point of contact and the point of intersection of tangent and $x \text {-axis }=2 x$

To find: we have to find the equation of curve which passes through the point $\left ( 1,2 \right )$

Hint: Use equation of tangent at  $P(x, y) \text { is }(Y-y)=\frac{d y}{d x}(X-x)$

Solution: Equation of tangent at $P(x,y)$

Put $Y=0$

\begin{aligned} &=-y=\frac{d y}{d x}(X-x) \\\\ &=-y \frac{d x}{d y}=X-x \\\\ &=X=x-y \frac{d x}{d y} \end{aligned}

Let co-ordinate of   $B=\left(x-y \frac{d x}{d y}, 0\right)$

Given distance between foot of ordinate of the point of contact and the point of intersection of tangent and $\mathrm{x} \text {-axis }=2 x \text { i.e. } B C=2 x$

$=\sqrt{\left(x-y \frac{d x}{d y}-x\right)^{2}+(0)^{2}}=2 x$            [Distance formula  $=\sqrt{\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}}$]

\begin{aligned} &=\sqrt{(-1)^{2}\left(y \frac{d x}{d y}\right)^{2}}=2 x \\\\ &=\sqrt{1\left(y \frac{d x}{d y}\right)^{2}}=2 x \end{aligned}

\begin{aligned} &=y \frac{d x}{d y}=2 x \\\\ &=\frac{d x}{x}=2 \frac{d y}{y} \end{aligned}        [Separating variables]

Integration on both sides we get

$=\int \frac{dx}{x}=2 \int \frac{d y}{y}$

\begin{aligned} &=\log x=2 \log y+\log C \ldots(i) \end{aligned}

It passes through $\left ( 1,2 \right )$

$\begin{array}{ll} =\log 1=2 \log 2+\log C \\\\ =0=2 \log 2+\log C & {[\log 1=0]} \end{array}$

\begin{aligned} &=-2 \log 2=\log C \\\\ &=\log (2)^{-2}=\log C \quad\left[2 \log x=\log x^{2}\right] \\\\ &=\log \frac{1}{4}=\log C \end{aligned}

Using antilogarithm

$=C=\frac{1}{4}$

Substituting the value of C in equation (i) we get

\begin{aligned} &=\log x=2 \log y+\log \left(\frac{1}{4}\right) \\\\ &=\log x=\log y^{2}+\log \left(\frac{1}{4}\right) \end{aligned}

\begin{aligned} &=\log x=\log \frac{y^{2}}{4} \\\\ &=x=\frac{y^{2}}{4} \\\\ &=y^{2}=4 x \end{aligned}

Hence $y^{2}=4 x$ is the equation of the curve.